On 1/17/2012 5:08 AM, Bruno Marchal wrote:
SNIP
- I disagree that set theory might be more primitive than arithmetic.
Why? First because arithmetic has been proved more primitive than set
theory, and less primitive than logic. With logic we cannot define
numbers. with set, we can define numbers, even all of them (N, Z, Q,
... octonions, etc.). The natural numbers are often defined by the von
Neuman finite ordinal:
0 = { }
1 = {{{}}} = {0}
2 = {{}, {{}} } = {0, 1}
3 = {{}, {{}},{{}}},,{{} ,{{}} } } = {0, 1, 2}
...
n = {0, 1, 2, ..., n-1}
etc.
And you can define addition by the disjoint union cardinal, and
multiplication by the cardinal of cartesian product,
and then, you can *prove* the laws of addition and the laws of
multiplication. With arithmetic you cannot recover any axioms of set
theory, except for the hereditarily finite sets.
I am confused. It seems to me that you are admitting that sets are
more primitive than Arithmetic since what you wrote here is a
demonstration of how numbers supervene on set theoretic operations. The
fact that we can define the natural numbers via the von Neuman finite
ordinals is the equivalent of claiming that the natural numbers emerge
from the von Neuman finite ordinals (up to isomorphism!), so I am
confused by what you are claiming here! But whichever is the most
primitive, it is not more primitive than the neutral foundation of
existence in itself.
- As I said, I don't take the word "Existence" as a theory. I have no
clue what you mean by that. I was asking for a theory. You say that by
taking (N, +, *) as a primitive structure, I am no more neutral
monist, due to the use of + and *. This is not correct. It would make
neutral monism empty. We alway need ontological terms (here 0, s(0)
etc.) and laws relating those terms (here addition and multiplication).
No, I am not making Existence as a theory, it is merely a postulate
of my overall theory (if you can call what I have been discussing a
"theory"). I am using the notion ofExistence
<http://books.google.com/books?id=VttF6CuC-cQC&pg=PT170&lpg=PT170&dq=existence+Objectivist+epistemology&source=bl&ots=d2ZAMFVpJM&sig=CLaMS0Y9kVnB6UfbgwUsuCG3wsU&hl=en&sa=X&ei=vsMVT7WNM8nWtgfEvaTRAw&sqi=2&ved=0CFUQ6AEwBg#v=onepage&q=existence%20Objectivist%20epistemology&f=false>
as it is defined in Objectivist Epistemology. For example, as explained
in this video lecture:
http://www.youtube.com/watch?v=GfOS7xfxezA&feature=player_embedded
Neutral monist takes is empty in the sense that it shows the
coherent implication that the most basic ontological level cannot be
considered to have some definite set of properties to the exclusion of
others.
- All you arguments with the term "physical" are going through in
arithmetic, given that you agree that "physical" is not primitive. For
example, the physical world is not required to make sense of what is a
universal machine. It is required for human chatting on the net, but
such a physical world is provided by arithmetic. Including concurrency.
WE simply might have to agree to disagree.
- I don't do philosophy. I offer you a technical result only. I still
don't know if you grasped it, or if you have any problem with it.
You result has deep philosophical implications and as a student of
philosophy I am very interested in it.
If you agree to assume that the brain works like a material machine,
then arithmetic is enough and more than arithmetic is necessarily
useless: it can only make the mind body problem unecessarily more
complex. Primitive matter (time, space) becomes like invisible horse.
Not epiphenomena, but epinomena.
Again, we have to agree to disagree on this. The necessity of
physical implementation cannot be dismissed otherwise the scientific
method itself is empty and useless. Without the definiteness that the
physical world offers us is accepted there can be only idle speculation,
we saw this kind of thinking in the Scholastics
<http://en.wikipedia.org/wiki/Scholasticism> and know well how that was
such a terrible waste of time. So why are you advocating a return to
that? Ideal monism was pushed hard by Bishop Berkeley and failed back in
the 18th century, its flaw - that the material world becomes causally
ineffective epiphenomena - is not solved by your result, it is only more
explicitly shown. You seem to think that it is a virtue. No, sorry, it
is not a virtue for the simple reason that it makes the falsification of
the theory impossible thus rendering it useless as an explanation.
My alternative hypothesis has the chance of being falsified as it
predicts that the physical worlds actually observe must be representable
as Boolean algebras (up to isomorphisms).
Onward!
Stephen
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